Author: Jeff (Bubbles The Dev), Founder of FNBubbles420 Org
We present a novel theoretical investigation into the dynamics of a charged bead sliding frictionlessly on a circular loop that rotates about a vertical axis with a time-dependent angular velocity...
Many classical mechanics problems examine particle motion on rotating frames or in electromagnetic fields, but rarely both simultaneously with a time-dependent angular velocity...
\[ \mathcal{L} = \frac{1}{2}m R^2 \dot{\theta}^2 + \frac{1}{2}m \omega(t)^2 R^2 \sin^2\theta + \frac{1}{2}q B R^2 \dot{\theta} \sin\theta + mgR\cos\theta \]
\[ \frac{d}{dt} \left( \frac{\partial \mathcal{L}}{\partial \dot{\theta}} \right) - \frac{\partial \mathcal{L}}{\partial \theta} = 0 \]
\[ m R^2 \ddot{\theta} + \frac{1}{2}q B R^2 \cos\theta \cdot \dot{\theta} - m R^2 \omega(t)^2 \sin\theta \cos\theta - \frac{1}{2}q B R^2 \dot{\theta} \cos\theta + mgR\sin\theta = 0 \]
\[ \boxed{m R^2 \ddot{\theta} = m R^2 \omega(t)^2 \sin\theta \cos\theta - mgR\sin\theta} \]
A unique blend of classical and electromagnetic physics with time-varying mechanics, offering insights for simulation, education, and experimental research.
Keywords: Lagrangian, Physics, Rotation, Magnetic Field, Charged Particle Dynamics